Assumption REF  is a minimal requisite to apply Proposition REF .
The existing works [1]}, [2]} assume stronger conditions; the operator \(E_a\)  is assumed to be Hilbert-Schmidt, which implies the compactness.
[2]} assumes that
the joint distribution of \(W\)  and \(Z\)  conditioned on \(a\)  is absolutely
continuous with respect to the product measure of \(P_{W|A=a}\)  and
\(P_{Z\vert A=a},\)  and its density is square integrable. The density
function serves as the integral kernel of the operator \(E_{a}\)  whose
\(L^{2}(P_{W|A=a}\otimes P_{Z\vert A=a})\) -norm corresponds to the
Hilbert-Schmidt norm of the operator, where we use \(\otimes \)  to denote the product measure.
On the other hand, in the setting of [1]}, all probability distributions have densities with respect to the Lebesgue measure. The operator \(E_a\)  (denoted by \(K_x\)  in [1]}) is defined by the relevant densities accordingly (see the paragraph after Lemma 2 of [1]}).
The compactness is, as in [2]}, established by the square-integrability of the integral kernel, which implies that the operator \(E_a\)  is Hilbert-Schmidt (see Condition A1 in [1]}).
